Rani started her journey from point 'P' and walked 10 km eastwards to reach…

In direction-test problems, each left or right turn rotates the direction of travel by 90 degrees from the current heading. Plotting every leg on a coordinate grid (East-West along one axis, North-South along the other) and comparing the horizontal and vertical displacement between two points tells you the direction of one point relative to the other.

1. Fix P at the origin, with East as the positive horizontal direction and North as the positive vertical direction.

2. P to Q: Rani walks 10 km East, so Q is 10 km east of P.

3. At Q she is facing East; turning left makes her face North. She walks 3 km North to reach R, so R is 3 km north of Q.

4. At R she is facing North; turning left makes her face West. She walks 12 km West to reach S, so S is 12 km west of R.

5. At S she is facing West; turning left makes her face South. She walks 3 km South to reach T, so T is 3 km south of S.

6. Relative to R: T is 12 km west of R (from the R-to-S leg) and 3 km south of R (from the S-to-T leg).

7. A point that lies simultaneously west and south of a reference point lies in the South-west direction from that point.

Independently: after R, the path turns west and then south without ever turning back east or north, so the net displacement from R must fall in the quadrant that is both west and south of R -- the South-west quadrant -- confirming the same result.